CONTROL CHARTS

Basic ConceptionsWhat is a control chart?The control chart is a graph used to study how a process changes over time. Data are plotted in time order. A control chart always has a central line for the average, an upper line for the upper control limit and a lower line for the lower control limit.

Lines are determined from historical data. By comparing current data to these lines

(http://www.asq.org/learn-about-quality/data-collection-analysis-tools/overview/control-chart.html)MANAGING FOR QUALITY AND PERFORMANCE EXCELLENCE, 7e, 2008 Thomson Higher Education Publishing

Examples of Control Charts

Basic Conceptions When to use a control chart?Controlling ongoing processes by finding and correcting problems as they occur. Predicting the expected range of outcomes from a process. Determining whether a process is stable (in statistical control). Determining whether the quality improvement project should aim to prevent specific problems or to make fundamental changes to the process.

Examples of Control Charts

Basic Conceptions Control Chart Basic Procedure Choose the appropriate control chart for the data.

Collect data, construct the chart and analyze the data. Look for out-of-control signals on the control chart. When one is identified, mark it on the chart and investigate the cause. Continue to plot data as they are generated. As each new data point is plotted, check for new out-of-control signals.

Basic Principles Basic components of control chartsA centerline, usually the mathematical average of all the samples plotted;Lower and upper control limits defining the constraints of common cause variations;Performance data plotted over time.

Important considerations in sampling for control charts 1.Choice of variable2.Sample size 3.Sampling frequency3a.To take large sized sample at less frequent interval3b.To take small sized samples at more frequent intervals.

Errors in drawing inferences from control charts1.type 1 error2.type 2 error

Basic PrinciplesGeneral model for a control chartUCL = + kCL = LCL = k where is the mean of the variable, and is the standard deviationof the variable. UCL=upper control limit; LCL = lower control limit; CL = center line. where k is the distance of the control limits from the center line, expressed in terms of standard deviation units. When k is set to 3, we speak of 3-sigma control chart

Basic Principles of Control ChartsTypes of the control charts

Variables control chartsVariable data are measured on a continuous scale. For example: time, weight, distance or temperature can be measured in fractions or decimals. Applied to data with continuous distribution Attributes control chartsAttribute data are counted and cannot have fractions or decimals. Attribute data arise when you are determining only the presence or absence of something: For example, a report can have four errors or five errors, but it cannot have four and a half errors.Applied to data following discrete distribution

Basic Principles of Control ChartsVariables control charts X-bar and R chart (also called averages and range chart) moving averagemoving range chart (also called MAMR chart) target charts (also called difference charts, deviation charts and nominal charts) CUSUM (cumulative sum chart) EWMA (exponentially weighted moving average chart) multivariate chart

Basic Principles of Control ChartsAttributes control charts p chart (proportion chart) np chartc chart (count chart) u chart

R-ChartAlways look at the Range chart first. The control limits on the X-bar chart are derived from the average range, so if the Range chart is out of control, then the control limits on the X-bar chart are meaningless.

Look for out of control points. If there are any, then the special causes must be eliminated.. Once the effect of the out of control points from the Range chart is removed, look at the X-bar Chart.

Example: R Control Chart

In the manufacturing of a certain machine part, the percentage of aluminum in the finished part is especially critical. For each production day, the aluminum percentage of five parts is measured. The table below consists of the average aluminum percentage of ten consecutive production days, along with the minimum and maximum sample values (aluminum percentage) for each day. The sum of the 10 samples means (below) is 258.8.

Day12345678910Sample Mean25.226.025.225.226.025.626.026.024.629.0Maximum Value26.627.627.727.427.627.427.527.926.831.6Minimum Value23.524.424.623.223.323.324.123.823.527.4

X-bar ChartThe X-bar chart monitors the process location over time, based on the average of a series of observations, called a subgroup. X-bar / Range charts are used when you can rationally collect measurements in groups (subgroups) of between two and ten observations.. For subgroup sizes greater than ten, use X-bar / Sigma charts,

For subgroup sizes equal to one, an Individual-X / Moving Range chart can be used, as well as EWMA or CuSum charts.

X-bar Charts are efficient at detecting relatively large shifts in the process average,

S ChartThe sample standard deviations are plotted in order to control the variability of a variable. For sample size (n>10), the S-chart is more efficient than R-chart.For situations where sample size exceeds 10, the X-bar chart and the S-chart should be used.

S**2 ChartIn this chart, the sample variances are plotted in order to control the variability of a variable.

Moving Average (MA)/Range Chart Moving Average / Range Charts are a set of control charts for variables data.

The Moving Average chart monitors the process location over time, based on the average of the current subgroup and one or more prior subgroups. The Range chart monitors the process variation over time

Always look at the Range chart first. The control limits on the Moving Average chart are derived from the average range, so if the Range chart is out of control, then the control limits on the Moving Average chart are meaningless.

Cumulative Sum (CUSUM) ChartThe CUSUM chart reacts more sensitively than the X-bar chart to a shifting of the mean value in the range of 0.5-2s; therefore, it is sited for monitoring processes with a high degree of imprecision.

This chart is particularly well-suited for detecting such small permanent shifts that may go undetected when using the X-bar chart.

Exponentially-weighted Moving Average (EWMA) ChartThe idea of moving averages of successive (adjacent) samples can be generalized. In principle, in order to detect a trend we need to weight successive samples to form a moving average; however, instead of a simple arithmetic moving average, we could compute a geometric moving average. It is also called Geometric Moving Average chart, see Montgomery, 1985, 1991).

EWMA Charts are generally used for detecting small shifts in the process mean.

EWMA Charts may also be preferred when the subgroups are of size n=1.

where is the weighting factor. The factor k is chosen generally to be 2 or 3.

Attributes Control Charts

An example of a common quality characteristic classification would be designating units as "conforming units" or "nonconforming units". Another quality characteristic criteria would be sorting units into "non defective" and "defective" categories. Quality characteristics of that type are called attributes. Examples of quality characteristics that are attributes are the number of failures in a production run, the proportion of malfunctioning wafers in a lot, the number of people eating in the cafeteria on a given day, etc. Control charts dealing with the number of defects or nonconformities are called c charts (for count). Control charts dealing with the proportion or fraction of defective product are called p chart (for proportion).

P-ChartTo evaluate process stability when counting the fraction defective. It is used when the sample size varies: the total number of circuit boards, meals, or bills delivered varies from one sampling period to the next.

Repeated samples of 150 coffee cans are inspected to determine whether a can is out of round or whether it contains leaks due to improper construction. Such a can is said to be nonconforming. Following is the data.

Sample12345678910Nonconforming#191046893104

np-ChartEvaluating process stability when counting the fraction defective. The np chart is useful when it's easy to count the number of defective items and the sample size is always the same. Examples might include: the number of defective circuit boards, meals in a restaurant, teller interactions in a bank, invoices, or bills. A fully capable process delivers zero defects. Although this may be difficult to achieve, it should still be our goal. Once we resolve the out-of-control point, we could use the quality problem solving process to begin to eliminate the common causes of defective paychecks. What are the most common types of paycheck errors? Why do they occur? What are the root causes of these paycheck errors?

C-ChartDetermining stability of "counted" data (e.g., errors per widget, inquiries per month, etc.) The c chart will help evaluate process stability when there can be more than one defect per unit. Examples might include: the number of defective elements on a circuit board, the number of defects in a dining experience--order wrong, food too cold, check wrong, or the number of defects in bank statement, invoice, or bill. This chart is especially useful when you want to know how many defects there are not just how many defective items there are. The c chart is useful when it's easy to count the number of defects and the sample size is always the same.

(http://www.qimacro

An automobile assembly worker is interested in monitoring and controlling the # of minor paint blemishes appearing on the outside door panel on the drivers side of a certain make of automobile. The following data were obtained, using a sample of 25 door panel.

Sample1234567----------25# of Paint Blemishes191046893-----------4

U-ChartDetermining stability of "counted" data (e.g., errors per widget, inquiries per month, etc.) when the sample size varies. The u chart will help evaluate process stability when there can be more than one defect per unit.

This chart is especially useful when you want to know how many defects there are not just how many defective items there are. It's one thing to know how many defective circuit boards, meals, statements, invoices, or bills there are; it is another thing to know how many defects were found in these defective items. It is used when the sample size varies: the number of circuit boards, meals, or bills delivered each day varies.

**With today's automated data collection systems, samples are frequently collected at closely spaced increments of time. Any sort of process dynamics introduces correlation into successive measurements, which causes havoc with standard control charts that assume independence between successive samples. In such cases, a control chart that captures the dynamics of the process must be used to properly detect unusual events when they occur.The proper chart for such situations is often an ARIMA control chart, which is based upon a parametric time series model for process dynamics. Such charts either plot the residual shocks to the system at each time period, or they display varying control limits based upon predicted values one period ahead in time.Control charts can also be used to monitor processes in which the mean measurement is expected to change over time. This commonly occurs when monitoring the wear on a tool, but also arises in other situations. The control charts for such cases have a centerline and control limits that follow the expected trend. One of the most important actions that can help maintain the quality of any good or service is to collect relevant data consistently over time, plot it, and examine the plots carefully. All statistical process control charts plot data (or a statistic calculated from data) versus time, with control limits designed to alert the analyst to events beyond normal sampling variability.STATGRAPHICS Centurion provides a very extensive collection of control charts. These include:1. Basic charts for variable data in which each point represents the most recent data, including X-Bar and R charts, X-Bar and S charts, X-Bar and S-squared charts, Median and Range charts, and Individuals charts based on X and MR(2).2. Basic charts for attribute data, including P, NP, U, and C charts.3. Time-weighted charts in which the points plotted are calculated from both current and historical data, including MA, EWMA, and CuSum charts.4. Multivariate charts, designed for situations where multiple correlated measurements are collected.5. ARIMA control charts for autocorrelated data in which the samples collected from one time period to the next are not independent.6. Toolwear charts for monitoring data that is expected to follow a trend line, not remain constant at a fixed level.7. Acceptance control charts for high Cpk processes, where the control limits are placed at a fixed distance from the specification limits rather than the centerline of the chart.8. CuScore charts, which are designed to detect specific types of patterns when they occur.All control charts can be used for Phase I studies, in which the data determine the location of the control limits, and Phase II studies, in which the data are compared against a pre-established standard. A special procedure is also provided to help design a control chart with acceptable power.Basic Variables ChartsThe classical type of control chart, originally developed back in the 1930's, is constructed by collecting data periodically and plotting it versus time. If more than one data value is collected at the same time, statistics such as the mean, range, median, or standard deviation are plotted. Control limits are added to the plot to signal unusually large deviations from the centerline, and run rules are employed to detect other unusual patterns. Basic Attributes ChartsFor attribute data, such as arise from PASS/FAIL testing, the charts used most often plot either rates or proportions. When the sample sizes vary, the control limits depend on the size of the samples. Time-Weighted ChartsWhen data is collected one sample at a time and plotted on an individuals chart, the control limits are usually quite wide, causing the chart to have poor power in detecting out-of-control situations. This can be remedied by plotting a weighted average or cumulative sum of the data, not just the most recent observation. The average run length of such charts is usually much less than that of a simple X chart. Multivariate Control ChartsWhen more than one variable are collected, separate control charts are frequently plotted for each variable. If the variables are correlated, this can lead to missed out-of-control signals. For such situations, STATGRAPHICS provides several types of multivariate control charts: T-Squared charts, Generalized Variance charts, and Multivariate EWMA charts. In the case of two variables, the points may be plotted on a control ellipse. ARIMA Control ChartsWith today's automated data collection systems, samples are frequently collected at closely spaced increments of time. Any sort of process dynamics introduces correlation into successive measurements, which causes havoc with standard control charts that assume independence between successive samples. In such cases, a control chart that captures the dynamics of the process must be used to properly detect unusual events when they occur.The proper chart for such situations is often an ARIMA control chart, which is based upon a parametric time series model for process dynamics. Such charts either plot the residual shocks to the system at each time period, or they display varying control limits based upon predicted values one period ahead in time. Toolwear ChartsControl charts can also be used to monitor processes in which the mean measurement is expected to change over time. This commonly occurs when monitoring the wear on a tool, but also arises in other situations. The control charts for such cases have a centerline and control limits that follow the expected trend. Acceptance Control ChartsFor processes with a high Cpk, requiring the measurements to remain within 3 sigma of the centerline may be unnecessarily restrictive. In such cases, the process may be allowed to drift, as long as it does not come too close to the specification limits. A useful type of control chart for this case is the Acceptance Control Chart, which positions the control limits based on the specification limits rather than the process mean.

Cuscore ChartsWhen monitoring a real-world process, the types of out-of-control situations that are likely to occur may be known ahead of time. For example, a pump that begins to fail may introduce an oscillation into the measurements at a specific frequency. In such cases, specialized CuScore Charts may be constructed to watch for that specific type of failure. STATGRAPHICS will construct CuScore charts to detect: spikes, ramps, bumps of known duration, step changes, exponential increases, sine waves with known frequency and phase, or any custom type of pattern that the user wishes to specify.*X-bar & Sigma charts are used when you can rationally collect measurements in groups (subgroups). Each subgroup represents a "snapshot" of the process at a given point in time. The charts' x-axes are time based, so that the charts show a history of the process. For this reason, you must have data that is time-ordered, that is, entered in the sequence from which it was generated. If this is not the case, then trends or shifts in the process may not be detected, but instead attributed to random (common cause) variation.*If 2 is the unknown variance of a probability distribution, then an unbiased estimator of 2 is the sample variance However, s, the sample standard deviation is not an unbiased estimator of . If the underlying distribution is normal, then s actually estimates c4 , where c4 is a constant that depends on the sample size n. This constant is tabulated in most text books on statistical quality control and may be calculated using C4 factor

*Variable control charts can by constructed for individual observations taken from the production line, rather than samples of observations. This is sometimes necessary when testing samples of multiple observations would be too expensive, inconvenient, or impossible. For example, the number of customer complaints or product returns may only be available on a monthly basis; yet, one would like to chart those numbers to detect quality problems. Another common application of these charts occurs in cases when automated testing devices inspect every single unit that is produced. In that case, one is often primarily interested in detecting small shifts in the product quality (for example, gradual deterioration of quality due to machine wear). The CUSUM, MA, and EWMA charts of cumulative sums and weighted averages discussed below may be most applicable in those situations. Moving Average / Range Charts are a set of control charts for variables data (data that is both quantitative and continuous in measurement, such as a measured dimension or time). The Moving Average chart monitors the process location over time, based on the average of the current subgroup and one or more prior subgroups. The Range chart monitors the process variation over time. The plotted points for a Moving Average / Range Chart, called a cell, include the current subgroup and one or more prior subgroups. Each subgroup within a "cell" may contain one or more observations, but must all be of the same size. Moving Average Charts are generally used for detecting small shifts in the process mean. They will detect shifts of .5 sigma to 2 sigma much faster than Shewhart charts with the same sample size. They are, however, slower in detecting large shifts in the process mean. In addition, typical run tests cannot be used because of the dependence of data points. Always look at the Range chart first. The control limits on the Moving Average chart are derived from the average range, so if the Range chart is out of control, then the control limits on the Moving Average chart are meaningless. W is the moving (cell) size.

*Variable control charts can by constructed for individual observations taken from the production line, rather than samples of observations. This is sometimes necessary when testing samples of multiple observations would be too expensive, inconvenient, or impossible. For example, the number of customer complaints or product returns may only be available on a monthly basis; yet, one would like to chart those numbers to detect quality problems. Another common application of these charts occurs in cases when automated testing devices inspect every single unit that is produced. In that case, one is often primarily interested in detecting small shifts in the product quality (for example, gradual deterioration of quality due to machine wear). The CUSUM, MA, and EWMA charts of cumulative sums and weighted averages discussed below may be most applicable in those situations. To establish control limits in such plots, Barnhard (1959) proposed the so-called V- mask, which is plotted after the last sample (on the right). The V-mask can be thought of as the upper and lower control limits for the cumulative sums. However, rather than being parallel to the center line; these lines converge at a particular angle to the right, producing the appearance of a V rotated on its side. If the line representing the cumulative sum crosses either one of the two lines, the process is out of control.The CUSUM chart was first introduced by Page (1954); the mathematical principles involved in its construction are discussed in Ewan (1963), Johnson (1961), and Johnson and Leone (1962).

*Exponentially-weighted Moving Average (EWMA) Chart. The idea of moving averages of successive (adjacent) samples can be generalized. In principle, in order to detect a trend we need to weight successive samples to form a moving average; however, instead of a simple arithmetic moving average, we could compute a geometric moving average (this chart (see graph below) is also called Geometric Moving Average chart, see Montgomery, 1985, 1991). The advantage of Cusum, EWMA and Moving Average charts is that each plotted point includes several observations, so you can use the Central Limit Theorem to say that the average of the points (or the moving average in this case) is normally distributed and the control limits are clearly defined.

*To see the differences between various attribute charts, let's consider an example of the errors in an accounting process, where each month we process a certain number of transactions. The Np-Chart monitors the number of times a condition occurs, relative to a constant sample size, when each sample can either have this condition, or not have this condition. For our example, we would sample a set number of transactions each month from all the transactions that occurred, and from this sample count the number of transactions that had one or more errors. We would then track on the control chart the number of transactions with errors per month. The p-Chart monitors the percent of samples having the condition, relative to either a fixed or varying sample size, when each sample can either have this condition, or not have this condition. For our example, we might choose to look at all the transactions in the month (since that would vary from month to month), or a set number of samples, whichever we prefer. From this sample, we would count the number of transactions that had one or more errors. We would then track on the control chart the percent of transactions with errors per month. The c-Chart monitors the number of times a condition occurs, relative to a constant sample size. In this case, a given sample can have more than one instance of the condition, in which case we count all the times it occurs in the sample. For our example, we would sample a set number of transactions each month from all the transactions that occurred, and from this sample count the total number of errors in all the transactions. We would then track on the control chart the number of errors in all the sampled transactions per month. The u-Chart monitors the percent of samples having the condition, relative to either a fixed or varying sample size. In this case, a given sample can have more than one instance of the condition, in which case we count all the times it occurs in the sample. For our example, we might choose to look at all the transactions in the month (since that would vary month to month), or a set number of samples, whichever we prefer. From this sample, we count the total number of errors in all the transactions. We would then track on the control chart the number of errors per transactions per month.